Number Systems-NCERT Solutions

Class IX Math
NCERT Solutions For Real Numbers
Exercise– 1.1
1.  Is zero a rational number? Can you write it in the form where p and q are integers and q ≠ 0?
Ans. Yes, zero is a rational number. We can write it in the form
           
2.  Find six rational numbers between 3 and 4.
Ans. We know that there are an infinite number of rational numbers between two rational numbers.
           Therefore, six rational numbers between 3 and 4 can be:
           Let x = 3 and y = 4, also n = 6
                 
           (x + d), (x + 2d), (x + 3d), (x + 4d), (x + 5d) and (x + 6d).
           ⇒ The six rational numbers between 3 and 4 are:
           
3.  Find five rational numbers between
           ⇒ The Five rational numbers between ‘x’ and ‘y’ are:
           (x + d); (x + 2d); (x + 3d); (x + 4d)
           and (x + 5d).
           
4.  State whether the following statements are true or false. Give reasons for your answers.
       (i)   Every natural number is a whole number.
       (ii)   Every integer is a whole number                     (iii)       Every rational number is a whole number.
Ans. (i)   True statement
                    [ ⇒ The collection of all natural numbers             and 0 is called whole numbers]
       (ii)   False statement
                    [ ⇒ Integers such as –1, –2 are non-whole numbers]
       (iii)   False statement
                    [ ⇒ Rational number is not a whole number]
Exercise– 1.2
1.  State whether the following statements are true or false. Justify your answers.
      (i)   Every irrational number is a real number.
      (ii)   Every point on the number line is of the form where m is a natural number.
      (iii)   Every real number is an irrational number.
Ans. (i)   True statement, because all rational numbers and all irrational numbers form the group (collection) of real numbers.
       (ii)   False statement, because no negative number can be the square root of any natural number.
       (iii)   False statement, because rational numbers are also a part of real numbers.
2.  Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number:
Ans. No, if we take a positive integer, say 4 its square root is 2, which is a rational number.
           REMEMBER
           According to the Pythagoras theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
           In the figure:
           OB2 = OA2 + AB2
           
                                                                             
3.  Show how can be represented on the number line.
Ans. Let us take the horizontal line XOX’ as the x-axis. Mark O as its origin such that it represents 0.
           Cut off OA = 1 unit, AB = 1 unit.
           ⇒ OB = 2 units
           Draw a perpendicular
           Cut off BC = 1 unit.
                                                                             
           Since OBC is a right triangle.
           OB2 + OC2 = OC2
           22 + 12 = OC2 4 + 1 = OC2
           OC2 = 5
           
           With O as centre and OC as radius, draw an arc intersecting OX at D.
           Since OC = OD
           
Exercise– 1.3
1.  Write the following in decidmal form and say what kind of decimal expansion each has:
           
Ans.
       (i)   We have
              ⇒ The decimal expansion of is terminating.
       (ii)    Dividing 1 by 11, we have:
                     
Note:
               The bar above the digits indicates the block of digits that repeats. Here, the repeating block is 09.
                
                               
                Remainder = 0, means the process of division terminates.
                
                Thus, the decimal expansion is terminating.
       (iv)   Dividing 3 by 13, we have
                                                  
                Here, the repeating block of digits is 230769.
                
                Thus, the decimal expansion of is
                “non-terminating repeating”.
       (v)   Dividing 2 by 11, we have
                
                Here, the repeating block of digits is 18.
                
                Thus, the decimal expansion of is
                “non-terminating repeating”.
       (iv)   Dividing 329 by 400, we have
                                
                Remainder = 0, means the process of division terminates.
                
                Thus, the decimal expansion of is terminating.
2.  You know that Can you predict what the decimal expansions of are, without actually doing the long division? If so, how?
Ans.
           
           Thus, without actually doing the long division we can predict the decimanl expansions of the above given rational numbers.
3.  Express the following in the form
       where p and q are integers and q ≠ 0.
       
Ans.
       
       Since, there is one repeating digit.
       ⇒ We multiply both sides by 10,
       10x = (0.666...) × 10
       or 10x = 6.6666...
       ⇒ 10x – x = 6.6666... – 0.6666...
       or 9x = 6
       
       (ii)   
       and 100x = 47.777
       Subtracting (1) from (2), we have
       100x – 10x = (47.777. . .) &ndsah; (4.777. . .) 90x = 43
       
       (iii)   
       Here, we have three repeating digits after the decimal point, therefore we multiply by 1000.
       
       → 1000x 0.001001 . . .
       or Subtracting (1) from (2), we have
       1000x – x = (1.001. . .) – (0.001. . .)
       or 999x = 1
       
4.  Let x = 0.99999 . . . in the form Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
Ans.
       Multiply both sides by 10, we have
       [⇒ There is only one repeating digit.]
       10× x = 10× (0.99999 . . .)
       or 9x = 9
5.  What can the maximum number of digits be in the repeating block of digits in the decimal expansion of Perform the division to check your answer.
Ans.
           Since, the number of entries in the repeating block of digits is less than the divisor. In the divisor is 17.
           ⇒ The maximum number of digits in the repeating block is 16. To perform the long division, we have
           
           The remainder 1 is the same digit from which we started the division.
           
           Thus, there are 16 digits in the repeating block in the decimal expansion of Hence, our answer is verified.
6.  Look at several examples of rational numbers in the form where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?
Ans.
           Let us look at decimal expansion of the following terminating rational numbers:
           
           We observe that the prime factorisation of q (i.e. denominator) has only powers of 2 or powers of 5 or powers of both.
           Note:
           If the denominator of a rational number
           (in its standard form) has prime factors either 2 or 5 or both, then and only then it can be represented as a terminating decimal.
7.  Write three numbers whose decimal expansions are non-terminating
           non-recurring.
Ans.
           
8.  Find three different irrational numbers between the rational numbers
Ans.
           To express decimal expansion of we have:
           
           As there are an infinite number of irrational numbers between any three of them can be:
           (i)      0.750750075000750. . .
           (ii)    0.767076700767000767. . .
           (iii)   0.78080078008000780. . .
9.  Classify the following numbers as rational or irrational:
       
Ans.
       (i)   ⇒23 is not a perfect square.
              ⇒ is an irrational number.
       (ii)   ⇒225 = 15 × 15 = 152
              ⇒225 is a perfect square.
              Thus, is a rational number.
       (iii)   ⇒0.3796 is a terminating decimal,
              ⇒It is a rational number.
       (iv)   
              Since, is a non-terminating and recurring (repeating) decimal.
              ⇒It is a rational number.
       (v)   Since, 1.101001000100001... is a non-terminating and non-repeating decimal number.
              ⇒It is an irrational number.
Exercise– 1.4
1.  Visualise 3.765 on the number line, using successive magnification.
Ans.
              3.765 lies between 3 and 4.
              Let us divide the interval (3, 4) into 10 equal parts.
        
              Since, 3.765 lies between 3.7 and 3.8 We again magnify the interval [3.7, 3.8] by dividing it further into 10 parts and concentrate the distance between 3.76 and 3.77.
              The number 3.765 lies between 3.76 and 3.77. Therefore we further magnify the interval [3.76, 3.77] into 10 equal parts.
              Now, the point corresponding to 3.765 is clearly located, as shown in Fig. (iii) above.
2.  Visualise 4.26 on the number line, up to 4 decimal places.
Ans.
              Note: We can magnify an interval endlessly using successive magnification.
              To visualize or 4.2626. . . on the number line up to 4 decimal places, we use the following steps.
               I.   The number 4.2626. . . lies between 4 and 5. Divide the interval [4, 5] in 10 smaller parts:
              
               II.   Obviously, the number 4.2626. . . lies between 4.2 and 4.3. We magnify the interval [4.2, 4.3]:
              
               III.   Next, we magnify the interval [4.26, 4.27]:
              
               IV.   Finally magnify the interval [4.262, 4263]:
              
              In Fig. (iv), we can easily observe the number 4.2626. . . or
Exercise– 1.5
1.  Classify the following numbers as rational or irrational:
       
Ans.
       (i)   
              Since it is a difference of a rational and irrational number,
               is an irrational number.
       (ii)   
              We have:
              
              which is a rational number.
       (iii)   
              Since,
               is a rational number.
       (iv)   
              ⇒ The quotient of rational and irrational is an irrational number.
               is an irrational number.
       (v)   2π
              ⇒ 2π = 2 × π = Product of a rational and an irrational (which is an irrational number)
              ⇒ 2π is an irrational number.
2.  Simplify each of the following expressions:
       
Ans.
       
3.  Recall,π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, This seems to contradict the fact that π is irrational. How will you resolve this contradiction?
Ans.
              When we measure the length of a line with a scale or with any other device, we only get an approximate rational value, i.e. c and d both are irrational.
              is irrational and hence π is irrational.
              Thus, there is no contradiction in saying that π is irrational.
4.  Rationalise the denominators of the following:
       
       
Ans.
       
       
Exercise– 1.6
1.  Find:
       (i)   641/2
       (ii)   321/5
       (iii)   1251/3
Ans.
       (i)   ⇒ 64 = 8 × 8 = 82
              ⇒ (64)1/2 = (82)1/2 = 82 × 1/2
                    [⇒ (am)n = am× n]
              ⇒ 81 = 8
              Thus, 641/2 = 8
       (ii)   ⇒ 32 = 2×2×2×2×2× = 25
              ⇒ (32)1/5 = (25)1/5 = 25×1/5
              [⇒(am)n = amn]
              = 21 = 2
              Thus, (32)1/5 = 2
       (iii)   ⇒ 125 = 5×5×5 = 53
              ⇒ (125)1/3 = (53)1/3 = 53×1/3 = 51
              Thus, 1251/3 = 5
2.  Find:
       (i)   93/2
       (ii)   322/5
       (iii)   163/4
       (iv)   125-1/3
Ans.
       (i)   ⇒ 9 = 3×3 = 32
              ⇒ (9)3/2 (32)3/2 = 32×3/2 = 33 = 27
              Thus, 93/2 = 27
       (ii)   ⇒ 32 = 2×2×2×2×2× = 25
              ⇒ (32)2/5 = (25)2/5 = 22 = 4
              Thus, 322/5 = 4
       (iii)   ⇒ 16 = 2×2×2×2 = 24
              ⇒ (16)3/4 = (24)3/4 = 24×3/4 = 23 = 8
              Thus, 163/4 = 8
       (iv)   ⇒ 125 = 5×5×5 = 53
              (125)-1/3 (53)-1/3
3.  Simplify:
     
Ans.
     
     
       (iv)   ⇒ am · bm = (ab)m
                ⇒ 71/2 · 81/2 = (7 × 8)1/2 = (56)1/2
                Thus, 71/2 · 81/2 = (56)1/2